Will my app sit on the Apple and Google stores under my own agency name or under somebody else's?

It can go up under your own agency name, and in most cases that is what you should insist on. When the app is published from an account that belongs to your business, the store listing shows your name as the seller, the reviews belong to you, and you keep control of the listing if you ever change the company that builds the app for you. If it is published from somebody else's account, your brand appears on the icon but the listing itself is not yours, and moving it later means starting again with a new app and no reviews.

Doing it your way means opening a developer account with Apple and with Google in your company name and giving your developer access to it. Both stores ask for proof that the business is real, so you will need your company registration details, and the checks take a little time. Whoever builds your app can walk you through it, but the account has to be created by you because it is tied to your legal entity.

Be ready for the review stage as well. Both stores read the listing and open the app before they let it go live, and a travel app that is only a wrapper around a website, or that has no working test login for the reviewer, is often sent back. That is normal and it is fixed by adding proper app behaviour and giving the reviewer a demo account.

Before you sign anything, ask in writing whose developer account the app will be published from, and make sure the answer is yours.